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Theorem opth 4237
Description: The ordered pair theorem. If two ordered pairs are equal, their first elements are equal and their second elements are equal. Exercise 6 of [TakeutiZaring] p. 16. Note that 𝐶 and 𝐷 are not required to be sets due our specific ordered pair definition. (Contributed by NM, 28-May-1995.)
Hypotheses
Ref Expression
opth1.1 𝐴 ∈ V
opth1.2 𝐵 ∈ V
Assertion
Ref Expression
opth (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝐴 = 𝐶𝐵 = 𝐷))

Proof of Theorem opth
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 opth1.1 . . . 4 𝐴 ∈ V
2 opth1.2 . . . 4 𝐵 ∈ V
31, 2opth1 4236 . . 3 (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → 𝐴 = 𝐶)
41, 2opi1 4232 . . . . . . 7 {𝐴} ∈ ⟨𝐴, 𝐵
5 id 19 . . . . . . 7 (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → ⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩)
64, 5eleqtrid 2266 . . . . . 6 (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → {𝐴} ∈ ⟨𝐶, 𝐷⟩)
7 oprcl 3802 . . . . . 6 ({𝐴} ∈ ⟨𝐶, 𝐷⟩ → (𝐶 ∈ V ∧ 𝐷 ∈ V))
86, 7syl 14 . . . . 5 (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → (𝐶 ∈ V ∧ 𝐷 ∈ V))
98simprd 114 . . . 4 (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → 𝐷 ∈ V)
103opeq1d 3784 . . . . . . . 8 (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → ⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐵⟩)
1110, 5eqtr3d 2212 . . . . . . 7 (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → ⟨𝐶, 𝐵⟩ = ⟨𝐶, 𝐷⟩)
128simpld 112 . . . . . . . 8 (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → 𝐶 ∈ V)
13 dfopg 3776 . . . . . . . 8 ((𝐶 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐶, 𝐵⟩ = {{𝐶}, {𝐶, 𝐵}})
1412, 2, 13sylancl 413 . . . . . . 7 (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → ⟨𝐶, 𝐵⟩ = {{𝐶}, {𝐶, 𝐵}})
1511, 14eqtr3d 2212 . . . . . 6 (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → ⟨𝐶, 𝐷⟩ = {{𝐶}, {𝐶, 𝐵}})
16 dfopg 3776 . . . . . . 7 ((𝐶 ∈ V ∧ 𝐷 ∈ V) → ⟨𝐶, 𝐷⟩ = {{𝐶}, {𝐶, 𝐷}})
178, 16syl 14 . . . . . 6 (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → ⟨𝐶, 𝐷⟩ = {{𝐶}, {𝐶, 𝐷}})
1815, 17eqtr3d 2212 . . . . 5 (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → {{𝐶}, {𝐶, 𝐵}} = {{𝐶}, {𝐶, 𝐷}})
19 prexg 4211 . . . . . . 7 ((𝐶 ∈ V ∧ 𝐵 ∈ V) → {𝐶, 𝐵} ∈ V)
2012, 2, 19sylancl 413 . . . . . 6 (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → {𝐶, 𝐵} ∈ V)
21 prexg 4211 . . . . . . 7 ((𝐶 ∈ V ∧ 𝐷 ∈ V) → {𝐶, 𝐷} ∈ V)
228, 21syl 14 . . . . . 6 (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → {𝐶, 𝐷} ∈ V)
23 preqr2g 3767 . . . . . 6 (({𝐶, 𝐵} ∈ V ∧ {𝐶, 𝐷} ∈ V) → ({{𝐶}, {𝐶, 𝐵}} = {{𝐶}, {𝐶, 𝐷}} → {𝐶, 𝐵} = {𝐶, 𝐷}))
2420, 22, 23syl2anc 411 . . . . 5 (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → ({{𝐶}, {𝐶, 𝐵}} = {{𝐶}, {𝐶, 𝐷}} → {𝐶, 𝐵} = {𝐶, 𝐷}))
2518, 24mpd 13 . . . 4 (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → {𝐶, 𝐵} = {𝐶, 𝐷})
26 preq2 3670 . . . . . . 7 (𝑥 = 𝐷 → {𝐶, 𝑥} = {𝐶, 𝐷})
2726eqeq2d 2189 . . . . . 6 (𝑥 = 𝐷 → ({𝐶, 𝐵} = {𝐶, 𝑥} ↔ {𝐶, 𝐵} = {𝐶, 𝐷}))
28 eqeq2 2187 . . . . . 6 (𝑥 = 𝐷 → (𝐵 = 𝑥𝐵 = 𝐷))
2927, 28imbi12d 234 . . . . 5 (𝑥 = 𝐷 → (({𝐶, 𝐵} = {𝐶, 𝑥} → 𝐵 = 𝑥) ↔ ({𝐶, 𝐵} = {𝐶, 𝐷} → 𝐵 = 𝐷)))
30 vex 2740 . . . . . 6 𝑥 ∈ V
312, 30preqr2 3769 . . . . 5 ({𝐶, 𝐵} = {𝐶, 𝑥} → 𝐵 = 𝑥)
3229, 31vtoclg 2797 . . . 4 (𝐷 ∈ V → ({𝐶, 𝐵} = {𝐶, 𝐷} → 𝐵 = 𝐷))
339, 25, 32sylc 62 . . 3 (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → 𝐵 = 𝐷)
343, 33jca 306 . 2 (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → (𝐴 = 𝐶𝐵 = 𝐷))
35 opeq12 3780 . 2 ((𝐴 = 𝐶𝐵 = 𝐷) → ⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩)
3634, 35impbii 126 1 (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝐴 = 𝐶𝐵 = 𝐷))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1353  wcel 2148  Vcvv 2737  {csn 3592  {cpr 3593  cop 3595
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601
This theorem is referenced by:  opthg  4238  otth2  4241  copsexg  4244  copsex4g  4247  opcom  4250  moop2  4251  opelopabsbALT  4259  opelopabsb  4260  ralxpf  4773  rexxpf  4774  cnvcnvsn  5105  funopg  5250  funinsn  5265  brabvv  5920  xpdom2  6830  xpf1o  6843  djuf1olem  7051  enq0ref  7431  enq0tr  7432  mulnnnq0  7448  eqresr  7834  cnref1o  9649  fisumcom2  11445  fprodcom2fi  11633  qredeu  12096  fnpr2ob  12758
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